Group (mathematics)

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A group is a mathematical structure consisting of set of elements combined with a binary operator which satisfies four conditions:

  1. Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
  2. Associativity: where , and are any element of the group
  3. Existence of Identity: there must exist an identity element such that ; that is, applying the binary operator to some element and the identity element leaves unchanged
  4. Existence of Inverse: for each element , there must exist an inverse such that

A group with commutative binary operator is known as Abelian.

Example: the Klein Four Group consists of the set of numbers {1, -1, i,-i} under the binary operation of multiplication, where i is the square root of -1, the basis of the imaginary numbers.

Groups are the appropriate mathematical structures for any application involving symmetry.